Average Error: 61.8 → 0.3
Time: 17.5s
Precision: 64
\[0.9000000000000000222044604925031308084726 \le t \le 1.100000000000000088817841970012523233891\]
\[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]
↓
\[t \cdot \left(t \cdot 3.999999999999999676487027278085939408227 \cdot 10^{-32}\right)\]
\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)
↓
t \cdot \left(t \cdot 3.999999999999999676487027278085939408227 \cdot 10^{-32}\right)
double f(double t) {
        double r3682755 = 1.0;
        double r3682756 = t;
        double r3682757 = 2e-16;
        double r3682758 = r3682756 * r3682757;
        double r3682759 = r3682755 + r3682758;
        double r3682760 = r3682759 * r3682759;
        double r3682761 = -1.0;
        double r3682762 = 2.0;
        double r3682763 = r3682762 * r3682758;
        double r3682764 = r3682761 - r3682763;
        double r3682765 = r3682760 + r3682764;
        return r3682765;
}
↓
double f(double t) {
        double r3682766 = t;
        double r3682767 = 3.9999999999999997e-32;
        double r3682768 = r3682766 * r3682767;
        double r3682769 = r3682766 * r3682768;
        return r3682769;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.8
Target50.6
Herbie0.3
\[\mathsf{fma}\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}, 1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}, -1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]

Derivation

  1. Initial program 61.8

    \[\left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 1.999999999999999958195573448069207123682 \cdot 10^{-16}\right)\right)\]

  2. Simplified50.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(t, 1.999999999999999958195573448069207123682 \cdot 10^{-16}, 1\right), \mathsf{fma}\left(t, 1.999999999999999958195573448069207123682 \cdot 10^{-16}, 1\right), -1 - \left(1.999999999999999958195573448069207123682 \cdot 10^{-16} \cdot t\right) \cdot 2\right)}\]

  3. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{3.999999999999999676487027278085939408227 \cdot 10^{-32} \cdot {t}^{2}}\]

  4. Simplified0.3

    \[\leadsto \color{blue}{\left(t \cdot 3.999999999999999676487027278085939408227 \cdot 10^{-32}\right) \cdot t}\]

  5. Final simplification0.3

    \[\leadsto t \cdot \left(t \cdot 3.999999999999999676487027278085939408227 \cdot 10^{-32}\right)\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (t)
  :name "fma_test1"
  :pre (<= 0.9 t 1.1)

  :herbie-target
  (fma (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16)) (- -1.0 (* 2.0 (* t 2e-16))))

  (+ (* (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16))) (- -1.0 (* 2.0 (* t 2e-16)))))